· 2011-03-29 · FIRST ORDER EIGENVALUE PERTURBATION THEORY AND THE NEWTON DIAGRAM Julio Moro Departamento de Matem´aticas Universidad Carlos III …

FIRST ORDER EIGENVALUEPERTURBATION THEORYAND THE NEWTON DIAGRAM

Julio MoroDepartamento de MatematicasUniversidad Carlos III de [email protected]

Froilan M. DopicoDepartamento de MatematicasUniversidad Carlos III de [email protected]

Abstract First order perturbation theory for eigenvalues of arbitrary matrices issystematically developed in all its generality with the aid of the Newtondiagram, an elementary geometric construction first proposed by IsaacNewton. In its simplest form, a square matrix A with known Jordancanonical form is linearly perturbed to A(ε) = A+ ε B for an arbitraryperturbation matrix B, and one is interested in the leading term in theε-expansion of the eigenvalues of A(ε). The perturbation of singularvalues and of generalized eigenvalues is also covered.

Keywords: eigenvalues, singular values, generalized eigenvalue problems, perturba-tion theory, asymptotic expansions, Newton diagram, Jordan canonicalform.

1. IntroductionEigenvalue perturbation theory has been an issue in applied mathe-

matics since Lord Rayleigh laid its foundations in [25]. One of his calcu-lations aimed at determining both the eigenfrequencies and eigenmodesof an oscillatory string with constant elasticity modulus and whose massdensity was a small deviation of a constant value. This particular prob-

1

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lem illustrates perfectly the typical setting of eigenvalue perturbationtheory: the matrix or operator under study is assumed to be a slightdeviation from some close, simpler matrix or operator for which thespectral problem is completely (and, in most cases, easily) solved. Thegiven operator A is replaced by a neighboring operator A whose eigen-values and eigenvectors are known. Then, the influence of the differenceB = A − A on the spectral objects is analyzed using an appropriateperturbation theory. The usual approach is introducing a perturbationparameter ε and considering the (matrix or) operator

A(ε) = A + εB. (1)

Its eigenvalues and eigenvectors depend on ε and are assumed to con-verge to the corresponding eigenvalues and eigenvectors as ε goes tozero. In a first stage of the analysis the leading terms of the ε-expansionsof the spectral objects of A(ε) are determined (hence the name of firstorder perturbation theory). In a second stage the convergence of theseexpansions is justified up to ε = 1. This paper shall be mainly focusedon the first stage, although some mention will be made of the appropriateconvergence results.

This general framework owes much to the formalism proposed bySchrodinger [33] in his approach to quantum mechanics. He describedany observable of a quantum mechanical system as a selfadjoint opera-tor H (the Hamiltonian in the case of the energy) defined in a certainHilbert space. An isolated eigenvalue λ0 and its corresponding normal-ized eigenvector x0 are interpreted as a bound state x0 with energylevel λ0. If the system is influenced by some external field, or if somepreviously neglected interaction has to be taken into account, a pertur-bation H1 must be added to the operator H. The question arises ofwhether there is at least one bound state in the neighborhood of the un-perturbed one. One also wants to calculate the bound states and energylevels of H +H1 in the neighborhood of x0, λ0. Schrodinger was one ofthe first to take the approach (1), considering H + εH1 and postulatingthe analytic dependence of the perturbed spectral objects on the pertur-bation parameter ε. The explicit formulas he obtained for eigenvaluesand eigenvectors are known as perturbation series in quantum mechan-ics. Even the splitting of multiple eigenvalues was studied in the firstorder approximation.

Schrodinger, however, did not prove the convergence of such expan-sions. That issue was finally settled for isolated eigenvalues of selfadjointoperators in Hilbert space by Rellich in a series of papers [26, 27, 28, 29,30] which stimulated further advances in similar problems. On one hand,Sz.-Nagy [38] translated Rellich’s arguments into the complex domain

First Order Eigenvalue Perturbation Theory and the Newton Diagram 3

using the Cauchy-Riesz integral method. This led, in particular, to theextension of Rellich’s results to the case of nonselfadjoint operators. Onthe other hand, Friedrichs [4] developed the perturbation theory of con-tinuous spectra, which turned out to be extremely useful in scatteringtheory and quantum field theory. Meanwhile, the perturbation theoryfor one-parameter semigroups was developed by Hille and Phillips [23, 6],and a general framework for the perturbation theory of linear operatorswas presented by Kato in his well-known treatise [9].

As can be seen from this discussion of the early history of the sub-ject, a great deal of effort was invested in developing eigenvalue pertur-bation theory for infinite-dimensional operators. In other words, firstorder eigenvalue perturbation theory began with a strong flavor of func-tional analysis. The essential ingredients, however, are purely finite-dimensional and can be presented in a strictly finite-dimensional settingwithout losing any of the mathematical subtleties of the original prob-lem1. This is precisely the main goal of the present paper: presenting, inan accessible way and without much technical apparatus, the essentialsof first order eigenvalue perturbation theory. It is also our intention toshow in § 3.2 that the general problem of determining the leading termof the ε-expansion of the eigenvalues of (1) as a function of B is farfrom being completely solved, even in the finite-dimensional case.

We choose not to present eigenvector perturbation results for two mainreasons: the first one is to keep the presentation relatively concise. Thesecond, and more important one, is that the techniques we will employare able to produce such kind of results only indirectly, and under quiterestrictive assumptions.

Making use of a powerful algebraic tool like the Newton diagrammethod will allow us to present a systematic development of the rele-vant results by means of purely matrix-analytic techniques. The resultswe will present are chiefly those obtained by Vishik and Lyusternik [41]and Lidskii [14], together with some consequences and extensions which,to our knowledge, are new (see § 3.2.1 and § 4 below). The results in[41], intended to be applied on differential operators, were generalized byLidskii [14] for the finite-dimensional case. He obtained simple explicitformulas for the perturbation coefficients and provided, at the same time,a much more elementary proof. The results in both [41] and [14] werelater refined by Baumgartel (see [1] § 7.4), in the sense of dealing notonly with perturbation series for eigenvalues and eigenvectors, but alsowith the corresponding eigenprojections as functions of ε. Vainberg and

1Actually, the original problem can be reduced to a purely finite-dimensional one via theresolvent integration method (see [1, chapter 10])

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Trenogin ( [39] § 32), on the other hand, offer a fairly thorough accountof similar results, obtained for Fredholm operators by applying the tech-niques of branching theory. Langer and Najman [10, 11, 12] generalizedLidskii’s results to analytic matrix functions A(λ) + B(λ, ε), using thelocal Smith normal form of parameter-dependent matrices (see section5 for more references).

From the functional analytic bias mentioned above one might getthe impression that first order eigenvalue perturbation theory is suitedmostly for dealing with infinite-dimensional operators, the finite-dimen-sional results being just a simple by-product of the former. Also, numer-ical analysts have not been very keen on first order eigenvalue perturba-tion results or, as they are also known, on perturbation expansions. Thatis partly true, mainly because perturbation bounds, as opposed to per-turbation expansions, are more amenable to backward rounding-erroranalysis, the most widespread tool for analyzing the numerical stabil-ity of algorithms for spectral problems. However, first order eigenvalueperturbation results are still extremely useful for the numerical analystwhenever some qualitative information is needed, e.g. on the splittingdirections of a multiple eigenvalue, even if perturbation bounds are avail-able (which is not always the case). Borrowing the very words of Stewartand Sun in their book [35, p. 292],

Although these [perturbation] expansions are usually corollaries ofmore general results, in many cases the general results themselves wereconjectured by looking at first order expansions. The reason is that theexpansions often tell ninety percent of the story and yet are free of theclutter that accompanies rigorous upper bounds.

Thus, first order eigenvalue perturbation theory has a role to playin applied mathematics, as well as a definite place among the manydifferent tools-of-the-trade of the applied mathematician.

The paper is organized as follows: the Newton diagram technique ispresented in detail in section 2, together with some elementary examples.Section 3 contains the results on perturbation of the standard eigenvalueproblem: after setting the stage for using the Newton diagram in thiscontext, the main result (Theorem 2) is proved and discussed in § 3.1.The cases outside the scope of Theorem 2 (what we call the nongenericcase) are analyzed in § 3.2. Particular attention is given in § 3.2.1 tothe case when the perturbation B has small rank. Section 4 presentsfirst order perturbation results for singular values, which can be easilyrecovered from the results in section 3. The final section 5 is devotedto a brief review of some of the available results for the perturbation ofgeneralized eigenvalue problems.


2. The Newton DiagramAlthough the Newton diagram (also called Newton polygon, or Newton-

Puiseux diagram) can be applied in much more general settings (in-cluding analytic perturbation of infinite-dimensional operators, see [2,§ III.8.3], [39, § 1.2] or [1, Appendix A7]), we strictly confine our presen-tation to the particular context we are dealing with, namely eigenvaluesas roots of the characteristic polynomial of a parameter-dependent ma-trix A(ε) = A+εB. Therefore, consider a complex polynomial equation

P (λ, ε) = λn + α1(ε)λn−1 + . . . + αn−1(ε)λ + αn(ε) = 0 (2)

in λ, with analytic coefficients

αk(ε) = αkεak + . . . , k = 1, . . . , n, (3)

where ak is the leading exponent and αk the leading coefficient of αk(ε)(i.e. αk 6= 0 and no term of order lower than ak appears in the expan-sion of αk(·)). For our convenience we set α0(ε) ≡ 1, i.e. α0 = 1 anda0 = 0.

It is well known [1, 9] that the roots λ of (2) are given by expansionsin fractional powers of ε. Our goal is to determine both the leadingexponents and the leading coefficients of these ε-expansions. To do that,we make the Ansatz

λ(ε) = µεβ + . . . , (4)

with µ, β to be determined. Substituting the Ansatz into (2), eachαk(ε)λn−k produces a term of order εak+(n−k)β plus higher order terms.Hence, P (λ, ε) is an infinite sum of powers of ε, each of them multipliedby an ε-independent coefficient. If, as we assume, λ(ε) is a root of (2),then all the coefficients must be zero. In particular, the lowest order in εmust be present at least twice among the exponents {ak +(n−k)β}n

k=0,i.e. there exist at least two indices i, j ∈ {0, 1, . . . , n} such that

ai + (n− i)β = aj + (n− j)β ≤ ak + (n− k)β, k = 0, 1, . . . , n.

In order to interpret geometrically this inequality, we plot the values ak

versus k for k = 0, 1, . . . , n on a cartesian grid. Then, the segment Sjoining (i, ai) with (j, aj) has slope β, and, since εai+(n−i)β is the termwith the lowest order, no other point (k, ak) lies below the straight linecontaining S. Therefore, if we draw the lower boundary of the convexhull of all the points {(k, ak)}n

k=0, it is clear that S must be on thatboundary. Thus, the slopes of the segments on the lower boundary arejust the exponents β appearing in (4). By imposing for each β that the

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coefficient of εai+(n−i)β be zero we obtain that the leading coefficients µof the eigenvalues of order εβ are the solutions of

∑

k∈Is

µn−kαk = 0, (5)

where the set Is = {k : (k, ak) ∈ S} may contain indices other thani, j (e.g. see Fig. 1(a) below, where three points (k, ak) lie on S).If we denote kmax = max Is and kmin = min Is, then the polynomialequation (5) has n−kmax zero roots, corresponding to the roots of orderhigher than β, and kmax − kmin nonzero roots which are the leadingcoefficients we are looking for. As a consequence, the number of roots of(2) with leading exponent β is given by the length of the projection onthe horizontal axis of the segment S. Notice that the total sum of thelengths of these projections is n.

Thus, we have derived a quite simple geometrical construction, goingback to Newton, which very simply provides us with all the leading pow-ers and leading coefficients we were looking for. The crucial ingredientis the so-called Newton diagram or Newton polygon.

Definition of the Newton diagram: Given a polynomial equation ofthe form (2) with analytic coefficients given by (3), plot ak versus k fork = 0, 1, . . . , n (if αk(·) ≡ 0, the corresponding point is disregarded).Denote each of these points by πk = (k, ak) and let

Π = {πk : αk 6≡ 0}

be the set of all plotted points. Then, the Newton diagram associatedwith P (λ, ε) is the lower boundary of the convex hull of the set Π.

With this terminology, the procedure outlined above for determiningthe leading terms of the asymptotic expansions (4) goes as follows:

Newton diagram procedure: Given a polynomial (2) with analyticcoefficients given by (3),

1) draw the associated Newton diagram.

2) The leading exponents β of the roots of (2) are the different slopesof the segments forming the Newton diagram.

3) The number of roots of order εβ is given by the length of theprojection on the horizontal axis of the segment with slope β.


4) The leading coefficients µ for each root of order εβ are the nonzeroroots of equation (5), where S is the segment of the Newton dia-gram with slope β.

We illustrate the procedure with two specific examples, depicted inFigure 1 below.

(a) (b)

Figure 1. Newton diagrams associated with the polynomials: a) λ5 + (2ε2 − ε3)λ4 −ελ3 +(−6ε2 +3ε5)λ+ε3−ε4; and b) λ4− (ε+2ε2)λ3 +(ε2−1)λ2 +(ε2−ε3)λ+2ε2

Example 1. Let P1(λ, ε) = λ5+(2ε2−ε3)λ4−ελ3+(−6ε2+3ε5)λ+ε3−ε4. Then, Π = {π0, π1, π2, π4, π5} and the Newton diagram associatedwith P1 is the one in Figure 1(a). It consists of two segments, one ofslope 1/2 connecting the points π0, π2, π4, and one of slope 1 connect-ing π4 with π5. Therefore, P1 has roots of orders

√ε and ε. More

precisely, there are four roots of order√

ε whose leading coefficients arethe nonzero roots of

α0µ5 + α2µ

3 + α4µ = µ5 − µ3 − 6µ = µ(µ2 − 3)(µ2 + 2),

i.e ±√3 and ± i√

2. Finally, there is one root of order ε whose leadingcoefficient is 1/6, the only root of

α4µ + α5 = −6µ + 1.

Example 2: Let P2(λ, ε) = λ4−(ε+2ε2)λ3+(ε2−1)λ2+(ε2−ε3)λ+2ε2.Now Π = {πi}4

i=0. The corresponding Newton diagram is drawn inFigure 1(b). Hence, there are two roots of O(1) with leading coefficients±1, the nonzero roots of

µ4 + α2µ2 = µ4 − µ2,

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and two roots of O(ε), whose leading coefficients ±√2 solve

α2µ2 + α4 = −µ2 + 2 = 0.

We stress that the whole argument above, leading to the Newton di-agram technique relies completely on the initial assumption that a con-vergent asymptotic expansion (4) exists. Otherwise, we might be com-puting the leading term of a nonexisting quantity. Of course, that wasnot a concern for Newton: he proposed this method in letters to Leibnizand Oldenburg (see [2, pp. 372-375], [21, pp. 20-42]), and developed itlater in his treatises [22], handling infinite series but saying nothing asto their convergence. Only in the 19th century, Puiseux [24] proved,in the course of his investigations on singularities, that the expansionsobtained through the Newton diagram converge in a neighborhood ofε = 0. Hence the name of Newton-Puiseux diagram.

Once this tool is at our disposal, we set to the task of obtaining firstorder results on perturbation of matrix eigenvalues

3. First order perturbation bounds for thestandard eigenvalue problem

Let λ0 be an eigenvalue of algebraic multiplicity a of the complexmatrix A ∈ Cn×n, and consider a perturbation

A(ε) = A + εB

for arbitrary B ∈ Cn×n. It is a well known fact [1, 9] that, for sufficientlysmall ε, the matrix A(ε) has a eigenvalues λj(ε) with λj(0) = λ0,each of them admitting an expansion in fractional powers of ε. Ourgoal is to determine the leading term of each expansion applying theNewton diagram technique to the characteristic polynomial P (λ, ε) =det(λI −A− ε B) of A(ε).

In order to prove our main perturbation result (Theorem 2 below),it is crucial to carefully determine which points (k, ak) may appearon the Newton diagram for a particular given Jordan structure of theunperturbed matrix A: let

J

J

=

Q

Q

A

[P P

](6)

be a Jordan decomposition of A, i.e.

Q

Q

[

P P]

= I. (7)


The matrix J contains all Jordan blocks associated with the eigenvalueof interest λ0, while J is the part of the Jordan form containing theother eigenvalues. Let

J = Γ11 ⊕ . . .⊕ Γr1

1 ⊕ . . .⊕ Γ1q ⊕ . . .⊕ Γrq

q , (8)where, for j = 1, . . . , q,

Γ1j = . . . = Γrj

j =

λ0 1· ·· ·· 1

λ0

is a Jordan block of dimension nj , repeated rj times, and ordered sothat

n1 > n2 > . . . > nq.

The nj are called the partial multiplicities for λ0. The eigenvalue λ0 issemisimple (nondefective) if q = n1 = 1 and nonderogatory if q = r1 =1. The algebraic and geometric multiplicities of λ0 are, respectively,

a =q∑

j=1

rjnj and g =q∑

j=1

rj . (9)

We further partition

P =

P 1

1 . . . P r11 . . . P 1

q . . . Prqq

conformally with (8). The columns of each P kj form a right Jordan

chain of A with length nj corresponding to λ0. If we denote by xkj the

first column of P kj , each xk

j is a right eigenvector of A associated withλ0. Analogously, we split

Q =

Q11...

Qr11...

Q1q...

Qrqq

,

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also conformally with (8). The rows of each Qkj form a left Jordan chain

of A of length nj corresponding to λ0. Hence, if we denote by ykj the

last (i.e. nj−th) row of Qkj , each yk

j is a left eigenvector correspondingto λ0. With these eigenvectors we build up matrices

Yj =

y1j...

yrj

j

, Xj = [x1

j , . . . , xrj

j ],

for j = 1, . . . , q,

Ws =

Y1...

Ys

, Zs = [X1, . . . , Xs],

for s = 1, . . . , q, and define square matrices Φs and Es of dimension

fs =s∑

j=1

rj (10)

by

Φs = WsBZs, s = 1, . . . , q,

E1 = I, Es =[

0 00 I

]for s = 2, . . . , q,

where the identity block in Es has dimension rs. Note that, due to thecumulative definitions of Ws and Zs, every Φs−1, s = 2, . . . , q, is theupper left block of Φs.

An important observation to be made at this point is that, althoughthe Jordan decomposition of A has been presented in its full generality,and all results presented below are valid for the general case (6), we mayassume with no loss of generality that λ0 is the only eigenvalue of A,i.e that J is empty. The reason is that, since we are only interestedin first order results, we may disregard quadratic terms in ε. Moreprecisely, if we write the characteristic polynomial of A(ε) as P (λ, ε) =det(λI − diag(J, J)− εB), with

B =

Q

Q

B

[P P

](11)


then one can use Schur’s formula to factorize P as

P (λ, ε) = det([

λI − J − εB11 −εB12

−εB21 λI − J − εB22

])= π(λ, ε) π(λ, ε),

withπ(λ, ε) = det(λI − J − εB22)π(λ, ε) = det(λI − J − εB11 − ε2S(λ, ε)),

where S is the matrix S(λ, ε) = B12(λI − J − εB22)−1B21. If λ is aneigenvalue of A(ε) close to λ0, then it cannot be a root of the polyno-mial π(·, ε), so it must be a root of the rational function π(·, ε), whichdepends on J only through terms of the second order in ε. Hence, Jhas no influence whatsoever on the first order terms and it is sufficient tostudy det(λI −J − εB11) to characterize the first order behavior. For arigorous proof, more formal than this plausibility argument, see Lidskii’soriginal paper [14, pp. 83-84] , or [1, § 3.9.1], where spectral projectionson the appropriate invariant subspaces are used to completely decoupleJ from the influence of J up to first order.

A second important simplification is that we may take λ0 = 0 atour convenience, since the shift A → A − λ0 I does not change eitherthe Jordan block structure or the Jordan chains of A. Hence, all resultsbelow are invariant under that transformation and λ0 may be set tozero.

With these simplifying assumptions, one can easily see that the partic-ular form of the leading terms of the eigenvalue expansions will dependmainly on the Jordan structure of A. To see it for the leading expo-nents, let J = P−1AP be a Jordan form of A, where we assume, asexplained above, that A has no eigenvalue other than λ0 = 0. In thatcase, the characteristic polynomial P (λ, ε) = det(λI−J − εB) of A(ε),with B = P−1BP, is a polynomial of the form (2). If the eigenvalueis semisimple, then J is zero and each αk(ε) equals εk multiplied bya certain sum of k-dimensional principal minors of B. In this case, theNewton diagram is formed by one single segment of slope s = 1. If theeigenvalue is not semisimple, some nontrivial Jordan block appears in J,so, besides the O(εk) terms, each αk(ε) contains lower order terms pro-duced by the −1s appearing above the diagonal of J . This clearly showsthat the effect of nontrivial Jordan blocks is to introduce in the Newtondiagram line segments with slopes less than 1. The smallest possibleslope corresponds to a nonderogatory eigenvalue (i.e. one single segmentof slope 1/n) and the largest possible one to the semisimple case. Allpossible Newton diagrams for the given multiplicity n lie between thesetwo extremal segments.

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This said, we are going to determine the lowest possible Newton dia-gram compatible with the given Jordan structure (8). We do it by fixingon the vertical axis an arbitrary height l, ranging between 1 and fq = g.For each height l we look for the rightmost possible point (k(l), l) whichmay be in the generating set Π of the Newton diagram for some suitableperturbation B. In other words, we are interested in

k(l) = max{k : ∃B ∈ Cn×n such that ak = l}as a function of l ∈ {1, . . . , fq}. The following theorem gives us thevalues of k(l) for the exponents l which are relevant to our argument(recall that fj is given by formula (10)).

Theorem 1 For every l ∈ {1, . . . , fq} the corresponding k(l) is equalto the sum of the dimensions of the l largest Jordan blocks of J. Moreprecisely, write f0 = 0 and suppose l = fj−1 + ρ, for some j = 1, . . . , qand 0 < ρ ≤ rj . Then,

k(l) = r1n1 + . . . + rj−1nj−1 + ρnj

and the coefficient of εl in αk(l) is equal to (−1)l multiplied by the sumof all principal minors of Φj corresponding to submatrices of dimensionl containing the upper left block Φj−1 of Φj (if j = 1, all principalminors of dimension l are to be considered). If, in particular, l =fj for some j ∈ {1, . . . , q}, then the coefficient of εfj in αk(fj) is(−1)fj detΦj .

Proof: As explained above, we may assume that λ0 = 0 is the onlyeigenvalue of the matrix A ∈ Cn×n. First, recall that given a n by nmatrix, the coefficient of λn−k in its characteristic polynomial is, exceptfor a sign, the sum of all k-dimensional principal minors of the matrix.In our case P (λ, ε) = det(λI − A(ε)) and each αk(ε) in (3) is, up toa sign factor, the sum of all k-dimensional principal minors of J + εB.Notice that the only elements of this matrix which are not of order εare the ones occupying the positions of the 1s above the diagonal of J .

Now, given l ∈ {1, . . . , fq}, we must determine the largest possiblek such that the sum of all k-dimensional principal minors is exactly oforder εl. In particular, the minor with the least order must be preciselyof that order, implying that the corresponding principal submatrix mustcontain k − l of the supradiagonal 1s. In other words, maximizing kfor the given l is equivalent to including as many 1s as possible in theprincipal submatrices, while still keeping the order εl. Clearly, this isachieved by choosing the rows and columns where the 1s are from the llargest Jordan blocks in J .


If l = fj for some j = 1, . . . , q, then there is only one way of choos-ing these blocks. Furthermore, one can easily check our claim on thecoefficient of εfj in αk(fj), since the leading term of αk(fj) is just thedeterminant of what is left from the k by k principal submatrix once therows and columns of the chosen k(fj)− fj supradiagonal 1s have beenremoved. The remaining matrix fj × fj matrix is precisely εΦj .

Finally, if l = fj + ρ with ρ < rj+1, there is more than one wayof choosing the l blocks: once the largest fj Jordan blocks have beenexhausted, each one corresponds to a different choice of ρ blocks amongthe rj+1 Jordan blocks of dimension nj+1. As to the coefficient of εl,the argument goes much in the same way as above.

As a consequence of Theorem 1, we conclude that the lowest possibleNewton diagram compatible with the Jordan structure (8) is the concate-nation of the segments Sj , j = 1, . . . , q connecting the points Pj−1 andPj , where Pj = (k(fj), fj) for each j = 1, . . . , q. This diagram, whichhas been called [16] the Newton envelope associated with the Jordanstructure (8), is not only the lowest possible, but also the most likelydiagram, since the actual diagram corresponding to a specific perturba-tion B will coincide with the Newton envelope unless some of the detΦj

vanishes, i.e. unless B satisfies an algebraic condition which confines itto an algebraic manifold (i.e. to a set of zero Lebesgue measure) in the setof matrices. In other words, the Newton envelope displays the genericbehavior of the eigenvalues of A under perturbation, in the sense thatit coincides with the Newton diagram in all situations except in thosenongeneric cases in which the perturbation B causes one of the Φj tobe singular. The next subsection is devoted to describe in detail thisgeneric behavior.

3.1 The generic case: Lidskii’s TheoremWe begin with the main result for generic perturbations, due to Lid-

skii [14] and, in a more restrictive version, to Vishik and Lyusternik [41].

Theorem 2 (Lidskii [14]) Let j ∈ {1, . . . , q} be given, and assumethat, if j > 1, Φj−1 is nonsingular. Then there are rjnj eigenvalues ofthe perturbed matrix A + εB admitting a first-order expansion

λklj (ε) = λ0 + (ξk

j )1/nj ε1/nj + o(ε1/nj ) (12)

for k = 1, . . . , rj , l = 1, . . . , nj , where

(i) the ξkj , k = 1, . . . , rj , are the roots of equation

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det (Φj − ξ Ej) = 0 (13)

or, equivalently, the eigenvalues of the Schur complement of Φj−1

in Φj (if j = 1, the ξk1 are just the r1 eigenvalues of Φ1);

(ii) the different values λklj (ε) for l = 1, . . . , nj are defined by taking

the nj distinct nj-th roots of ξkj .

If, in addition, the rj solutions ξkj of (13) are all distinct, then the

eigenvalues (12) can be expanded locally in power series of the form

λklj (ε) = λ0 + (ξk

j )1/nj ε1/nj +∞∑

s=2

akljs εs/nj , (14)

k = 1, . . . , rj , l = 1, . . . , nj .

Proof: Once the Newton diagram technique is at hand, the proof isa consequence of Theorem 1. First, suppose that both Φj−1 and Φj

are nonsingular. Then, both Pj−1 and Pj are in the set Π generatingthe Newton diagram, i.e. the segment Sj of slope 1/nj connecting bothpoints is one of the segments in the diagram (recall that no point (k, ak)can lie below Sj). This gives us the leading exponent of expansion (12).The leading coefficient comes from carefully examining equation (5).One can check that

∑

(k,ak)∈Sj

µn−kαk = µn−k(fj)

[µnjrj αk(fj−1) +

∑

t∈T

αk(fj−t)µt nj + αk(fj)

]= 0,

(15)where

T = {t ∈ {1, . . . , rj − 1} : Qt = (k(fj − t), fj − t) ∈ Π}, (16)

i.e. T is the set of indices corresponding to the intermediate pointsQt ∈ Π eventually lying on Sj . Notice that bracketed expression in(15) depends on µ only through µnj . Now, recall from Theorem 1 thatfor each l = fj − t with t ∈ T, the corresponding αk(l) is (up to thesign) the sum of all principal minors of Φj of dimension l containingΦj−1. This is precisely the way the coefficients of the powers of ξ areobtained in det(Φj − ξEj). Hence, the nonzero solutions of ( 15) aresolutions of det(Φj − µnjEj) = 0 as well.

Now, suppose that Φj is singular. Then, the corresponding pointPj no longer belongs to the diagram, implying the loss of some of the


expansions (12) or, equivalently, the loss of part of the segment Sj . Let βbe either β = rj if no point Qt = (k(fj−t), fj−t) is in Π or β = min Tfor T defined as in (16). In either case Qβ = (k(fj − β), fj − β) is therightmost point of Π on Sj . Then, the part of Sj which remains onthe Newton diagram is the segment connecting Pj−1 with Qβ. Thisaccounts for (rj − β)nj expansions (12), whose leading coefficients are,reasoning as above, the nj-th roots of the rj − β nonzero solutions ofequation (13). As to the βnj remaining eigenvalues, they correspond tosegments whose slope is strictly larger than 1/nj . Hence, expansion (12)is still valid, since they correspond to the β null solutions of equation(13).

As to expansion (14), one can check (see the original reference [14] or[16, § 2]) that, after the change of variables

z = ε1/nj

µ =λ

z

and an appropriate scaling in z, the polynomial equation P (λ, ε) =det(λI − A − εB) = P (µ, z) = 0 has, for any given z 6= 0, the sameroots as a new polynomial equation Q(µ, z) = 0 with

Q(µ, 0) = ±µα det(Φj − µnjEj)

for some suitable α ≥ 0. Hence, if all rj roots of equation (13) areknown to be distinct, the implicit function theorem can be applied toQ(µ, z) = 0, implying that the rjnj roots

µklj (z) = (ξk

j )1/nj + o(1), k = 1, . . . , rj ; l = 1, . . . , nj ,

of Q(µ(z), z) = 0 for small enough z are analytic functions of z = ε1/nj .

Two special cases of Theorem 2 are well known. If λ0 is semisimple,i.e. q = n1 = 1 with multiplicity r1, equation (12) reduces to

λk11 (ε) = λ0 + ξk

1 ε + o(ε), (17)

where the ξk1 are the eigenvalues of the r1 by r1 matrix Y1BX1 (cf. [9,

§ II.2.3]). On the other hand, if λ is nonderogatory, i.e. q = r1 = 1with multiplicity n1, equation (12) reduces to

λ1lj (ε) = λ0 + (ξ1

1)1/n1 ε1/n1 + o(ε1/n1),

where ξ11 = y1

1Bx11. These two cases coincide when λ is simple.

16

Theorem 2 does not address either the convergence or the ultimateform of the o(ε1/nj ) term in expansion (12), since these issues are beyondthe reach of a purely algebraic tool like the Newton diagram. One canshow by other means (see [1, § 9.3.1], [9, § II.1.2]) that whenever bothΦj−1 and Φj are nonsingular, the rjnj eigenvalues (12) acn be writtenas convergent power series in the variable ε1/nj . This is no longer true ifonly Φj−1 is nonsingular, unless some additional information, as in thelast part of Theorem 2, is available (see, for instance, the perturbationmatrix (18) in Example 3 below, for which det Φ2 = 0 and two out ofthe four eigenvalues corresponding to Φ2 are of order ε2/3)

However, one important special case deserves to be mentioned: ifboth the unperturbed matrix A and the perturbation matrix B arenormal, then all eigenvalues of A + εB are analytic functions of ε [1,§ 7.2], i.e. they have a convergent representation (14) with nj = 1. Thisproperty will be crucial in Section 4, when dealing with the perturbationof singular values.

We conclude this subsection by referring the reader interested in eigen-vector perturbation results to [14, Theorem 2], the eigenvector pertur-bation theorem in [14] analogous to the eigenvalue result above, whichessentially amounts to replacing (14) in the eigenvalue-eigenvector equa-tion A(ε)v(ε) = λ(ε)v(ε) (the same result appears in [16] as Theorem2.2).

3.2 Nongeneric perturbationsIf the perturbation is nongeneric, i.e. when the matrix B is such that

Φj−1 is singular, Theorem 2 does not apply. The question is what canwe say about the eigenvalues of A+εB in this case. The answer is: notmuch, at least in such a systematic way as in Theorem 2. Although theNewton diagram is still a powerful instrument which allows us to dealwith each particular case, it is not easy to give a clear, global picture ofthe wide variety of different possible behaviors. To give an idea of thedifficulties, consider the following example:

Example 3: Let A ∈ R8×8 be already in Jordan form,

A = J = Γ11 ⊕ Γ1

2 ⊕ Γ22 ⊕ Γ1

3

with n1 = 3, n2 = 2, n3 = 1 and r1 = r3 = 1, r2 = 2, i.e.


A =

0 1 00 0 10 0 0

0 10 0

0 10 0

0

.

According to Theorem 2, a generic perturbation, i.e. any 8 by 8 matrix

B =

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗¤ ∗ ∗ ♣ ∗ ♣ ∗ ♥∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗♣ ∗ ∗ ♣ ∗ ♣ ∗ ♥∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗♣ ∗ ∗ ♣ ∗ ♣ ∗ ♥♥ ∗ ∗ ♥ ∗ ♥ ∗ ♥

with all three submatrices

Φ1 =[

¤], Φ2 =

¤ ♣ ♣♣ ♣ ♣♣ ♣ ♣

, Φ3 =

¤ ♣ ♣ ♥♣ ♣ ♣ ♥♣ ♣ ♣ ♥♥ ♥ ♥ ♥

nonsingular gives rise to a Newton diagram coinciding with the Newtonenvelope. In that case, Π contains the points P0,P1,P2 and P3, withP0 = (0, 0), P1 = (3, 1), P2 = (7, 3) and P3 = (8, 4) (see Figure 3.2abelow). Hence, the eigenvalues of A split typically into three eigenvaluesof order ε1/3, four eigenvalues of order ε1/2 and one eigenvalue of orderε.

A good number of different possibilities arises whenever some of thematrices Φj turn out to be singular. As a first example, consider theperturbation matrix

B1 =

0 0 0 0 0 0 0 00 0 0 0 0 0 0 01 0 0 0 0 1 0 00 0 0 0 0 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 01 0 0 0 0 0 0 10 0 0 1 0 0 0 0

, (18)

18

for which

Φ3 =

1 0 1 00 0 1 01 0 0 10 1 0 0

.

Both Φ1 and Φ3 are invertible, but Φ2 is singular. In this case, the setΠ contains the points P0,P1 and P3, together with the point Q1 =(5, 2). According to the Newton diagram, shown in Figure 3.2b, thematrix A + εB1 has three eigenvalues of order ε1/3, two eigenvalues oforder ε1/2 and three eigenvalues of order ε2/3.

P1

3

2P

P

P0P0

Q1

P1

P3

Figure 2a. Newton diagram asso-ciated with the matrix A + ε B forgeneric B in Example 3.

Figure 2b. Newton diagrams associ-ated with the matrix A + ε B1 for thematrix B1 in (18). The Newton en-velope is shown as a dashed line, theNewton diagram as a solid one

Now, take the perturbation

B2 =

0 0 0 0 0 0 0 00 0 0 0 0 0 0 01 0 0 1 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 10 0 0 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 1 0 0

, (19)

producing

Φ3 =

1 1 0 00 0 0 10 1 0 00 0 1 0

.


Again, both Φ1 and Φ3 are invertible, but now neither P2 nor Q1 areon the Newton diagram (see Figure 2ba). Thus, the matrix A + εB2

has three eigenvalues of order ε1/3 and five eigenvalues of order ε3/5.If we take as our next nongeneric perturbation

B3 =

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 0 01 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 1

, (20)

so that

Φ3 =

0 1 0 01 0 0 00 0 1 00 0 0 1

,

then both Φ2 and Φ3 are invertible, but Φ1 is singular and the set Πcontains {P0,Q1,P2, P3}. The corresponding Newton diagram is theone in Figure 2bb, and A + εB3 has five eigenvalues of order ε2/5, twoeigenvalues of order ε1/2 and one eigenvalue of order ε.

P1

3

0P

P

P

3

2P

P

Q1

0

Figure 2a. Newton diagram associ-ated with A + ε B2 for the matrix B2

in (19). The Newton envelope is shownas a thin solid line, the Newton dia-gram as a thick one

Figure 2b. Newton diagram associ-ated with A + ε B3 for the matrix B3


20

Now, consider

B4 =

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 1 0 1 0 00 0 0 0 0 0 0 01 0 0 0 0 1 0 10 0 0 0 0 0 0 0−1 0 0 1 0 1 0 02 0 0 0 0 1 0 0

, (21)

with

Φ3 =

0 1 1 01 0 1 1−1 1 1 02 0 1 0

.

In this case, Φ1 is singular, while Φ2 and Φ3 are not. The point Q1

is not in the set Π and the Newton diagram is the one in Figure 2ba,so A + εB4 has seven eigenvalues of order ε3/7 and one eigenvalue oforder ε.

Finally, consider

B5 =

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 1 0 1 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 10 0 0 0 0 0 0 00 0 0 1 0 0 0 01 0 0 0 0 1 0 0

, (22)

with

Φ3 =

0 1 1 00 0 0 10 1 0 01 0 1 0

.

In this case, both Φ1 and Φ2 are singular and Π contains P0 and P3.Again, the point Q1 does not belong to Π, so the Newton diagram isa single segment of slope 1/2 (see Figure 2bb) and A + εB5 has eighteigenvalues of order ε1/2.

We conclude the example here, although the five instances above donot exhaust all possible behaviors: notice that Φ3 has been assumed


3

2P

P

P0 P0

3P

Figure 2a. Newton diagram associ-ated with A + ε B4 for the matrix B4


Figure 2b. Newton diagram associ-ated with A + ε B5 for the matrix B5


to be invertible in all four cases. Other possibilities arise if Φ3 be-comes singular, but we do not pursue the matter further for the sake ofbrevity. In any case, the example above displays the richness of differ-ent behaviors to be expected from nongeneric perturbations to matriceswith nontrivial Jordan forms.

In fact, the nongeneric case presents a very important additional dif-ficulty which is absent for the generic case: namely that one can nolonger make the simplifying assumption of disregarding the part J ofthe Jordan canonical form corresponding to eigenvalues other than λ0.The following example illustrates this:Example 4: Consider the matrix

A =

00

1

∈ R3×3, (23)

which is already in Jordan form, but having two different eigenvalues,and take the perturbation

B =

0 1 00 0 11 0 0

. (24)

Obviously, if we disregard the part corresponding to the nonzero eigen-

value 1, the perturbed matrix would reduce to[

0 ε0 0

]which still has

two zero eigenvalues. However, the characteristic polynomial of the 3 by3 matrix A + εB is P (λ, ε) = λ3−λ2− ε3. The corresponding Newton

22

diagram, in Figure 2b below, shows that A+ εB has one eigenvalues ofthe order of unity, and two eigenvalues of order ε3/2.

..

Figure 3. Newton diagram associated with the matrix A + ε B for the matricesdefined in (23) and (24).

Some attempts have been made to analyze the behavior for certainspecific classes of nongeneric perturbations. Some preliminary resultsmay be found in [16, § 3], but the first systematic description of struc-tured perturbations is obtained in Ma and Edelman [15] for upper k-Hessenberg perturbations of Jordan blocks. Following the ideas bothin [3] and [16], they show that if a n by n Jordan block is perturbed byan upper k-Hessenberg matrix (i.e. a matrix having the k subdiagonalsclosest to the main diagonal not zero), then the eigenvalue splits typicallyinto bn/kc eigenvalues of order ε1/k and, if k does not divide n, into radditional eigenvalues of order ε1/r, where r ≡ n mod k. The leadingcoefficients are also characterized. Again, as in Lidskii’s theorem, thetypical behavior (in this non-typical situation) is given by certain alge-braic conditions which in this case depend only on the elements of thek-th subdiagonal of B.

More recently, Jeannerod [8] has extended Lidskii’s results by ob-taining explicit formulas for both the leading exponents and leadingcoefficients of the Puiseux expansions of the eigenvalues of analytic per-turbations J + B(ε) of a Jordan matrix J, provided the powers of ε inthe perturbation matrix B(ε) conform in a certain way to the Jordanstructure given by J . To be more precise, the least power of ε in either ofany two columns of B corresponding to a same Jordan block of J mustbe the same. Lidskii’s Theorem 2 is recovered when that least power isε for all blocks, but certain nongeneric cases are covered in this moregeneral setting.

Nonetheless, in both cases [15, 8] the particular structure of the per-turbations to the Jordan blocks is not preserved by undoing the change


of basis leading to the Jordan form. Hence, not much information isprovided for nongeneric perturbations of arbitrary matrices. Therefore,it is safe to say that, so far, the problem of categorizing all possible be-haviors of the eigenvalues of A + εB as a function of the perturbationmatrix B is a very difficult open problem, which is still quite far frombeing solved.

However, there is still another specific class of nongeneric perturba-tions for which quite some information can be obtained

3.2.1 Low rank perturbations. A relevant class of nongenericperturbations are perturbations of low rank, which frequently arise whena system has to be controlled using less parameters than the numberof degrees of freedom defining the system [40]. We focus on how dothe eigenvalues and the Jordan structure of a matrix change when itis perturbed by a matrix of low rank, with particular attention to howmany Jordan blocks are destroyed for each eigenvalue.

Although some of the used techniques have a similar flavor to theNewton diagram technique, an important difference with the rest ofthe paper has to be highlighted: the perturbations studied here are notinfinitesimal. Thus we will be able to obtain absolute, instead of firstorder, results holding for perturbations of any magnitude. This said, webegin with an example which illustrates how a small rank perturbationtypically behaves.Example 5: Let us denote by Js(λ0) a Jordan block of dimension scorresponding to the eigenvalue λ0. Consider the following matrix

J = J3(1)⊕ J3(1)⊕ J3(1)⊕ J1(1)⊕ J3(−1)⊕ J2(−1)⊕ J2(−1)⊕J3(2)⊕ J3(2).

We have generated 200 random matrices B with rank 2 and computedthe Jordan canonical form 2 of A + B. The maximum and minimumvalues obtained for the ratios ‖B‖/‖J‖ have been 305 and 130, i.e. thematrices A+B are not by any means small perturbations of A. Surpris-ingly enough, in 171 out of the 200 cases analysed the Jordan canonicalform of A + B had one block J3(1), one block J1(1) and one blockJ2(−1). The rest of the blocks were one-dimensional blocks correspond-ing to eigenvalues different from 1,−1, 2. This experiment suggests the

2This was done with the command ’jordan’ of the Symbolic Math Toolbox in MATLAB5.3. To work properly this command requires the elements of the matrix argument to beintegers or ratios of small integers. Therefore, the random matrices B had moderately largeinteger elements, and are not as random as one might think. This is the reason why about15% of the generated matrices did not follow the generic behavior described in the example.

24

following generic behavior: If λ0 is an eigenvalue of A having g (geo-metric multiplicity) Jordan blocks and B is a matrix with rank m < g,then λ0 is also an eigenvalue of A + B having exactly g − m Jordanblocks equal to the g −m smallest Jordan blocks of λ0 in A, i.e., whenperturbing a matrix by a generic rank m perturbation, the m largestJordan blocks for each eigenvalue dissapear and the rest of the Jordanblocks remain. This “usual” behavior has been observed in many otherexperiments, but notice that it is easy to build m-rank perturbationswhich do not follow this rule. If m ≥ g, the eigenvalue λ0 is typicaly nolonger an eigenvalue of A + B, as expected.

Now we will try to understand this generic behavior. The next Lemmagives necessary and sufficient algebraic conditions for it.

Lemma 3 Let A be a matrix with Jordan form (6), i.e. having an eigen-value λ0 with Jordan blocks of dimensions n1 > n2 > . . . > nq repeatedr1, r2, . . . , rq times. Let m ∈ {1, . . . , g} be an integer number, whereg is the geometric multiplicity (9). Suppose m = fj−1 + ρ for somej = 1, . . . , q, 0 < ρ ≤ rj with f0 = 0. Then λ0 is an eigenvalue of thematrix A + B with Jordan blocks of dimensions nj > nj+1 . . . > nq re-peated rj−ρ, rj+1, . . . , rq times (if ρ = rj there are no blocks of dimensionnj) if and only if

a) rank(A + B−λ0I)k = rank(A−λ0I)k + k m, for all k = 1, 2, . . . , nj,and

b) rank(A + B − λ0I)k = rank(A + B − λ0I)nj for all k ≥ nj.

Proof: Just remember [7, pag. 127] that for an arbitrary matrix Cthe number dk(C, λ) = rank(C − λI)k−1 − rank(C − λI)k is the totalnumber of Jordan blocks of C of dimensions greater than or equal tok associated with the eigenvalue λ. Therefore dk(C, λ) − dk+1(C, λ)is the number of Jordan blocks of C associated to λ with dimensionequal to k. Thus Conditions a) and b) of the Lemma are equivalent todk(A + B, λ0) = dk(A, λ0)−m for k = 1, . . . , nj and dk(A + B, λ0) = 0for k > nj . The rest is trivial.

Notice that Lemma 3 does not force B to be a matrix of rank m. Infact, our goal is to justify why most matrices of rank m fulfill the previousequations a) and b) on the ranks. The following plausibility argumentshows, under mild restrictions on a matrix B with rank m = fj−1 + ρ,that something very unlikely has to happen for conditions a) and b)not to be fulfilled. In the first place, notice that for any k the matrix


(A− λ0I + B)k can be expanded as a sum of (A− λ0I)k plus productsof two matrices, with the left factor being of the form (A − λ0I)sBwith s ranging from 0 to k − 1. Secondly, elementary linear algebraresults show that if col(B) ∩ ker(A − λ0I)nj−1 = {0} (here col and kerstand for column and null spaces respectively) then rank(A− λ0I)sB =rank(B) = m for all s = 0, 1, . . . , nj − 1. Therefore, the equalitiesof item a) in Lemma 3 hold unless some “rank cancellation” occurs.It is well known that rank cancellations are very uncommon when theinvolved matrices have small rank, as in our case. With regard to theassumption col(B)∩ker(A−λ0I)nj−1 = {0}, it is really very mild, sincedim col(B) = m and dim ker(A − λ0I)nj−1 ≤ a −m. Hence, it followsthat the set of matrices B of rank m not satisfying this condition haszero Lebesgue measure.

No need to say that the previous argument does not prove any rigorousresult, but it sugests that the problem of understanding why the behaviordescribed in Example 5 is the most likely one is a problem of determiningthe Lebesgue measure of the set of matrices B of rank m not fulfillingthe conditions of Lemma 3. The most convenient way to deal withthis problem is to rewrite conditions a), b) in a simpler but equivalentform. All of these questions are beyond the scope of this survey becausedifferent techniques than the ones used here are required. However anargument much in the same spirit of the Newton diagram approachallows us to prove the following theorem which is an important steptowards the complete solution of this question. More on this will be saidin [17].

Theorem 4 Let A be a n× n matrix with Jordan form (6), i.e. havingan eigenvalue λ0 with Jordan blocks of dimensions n1 > n2 > . . . > nq

repeated r1, r2, . . . , rq times, and algebraic and geometric multiplicitiesgiven by (9). Let B be a n× n matrix with rank m = fj−1 + ρ for somej = 1, . . . , q, 0 < ρ ≤ rj with f0 = 0. Then the characteristic polynomialof A + B is of the form

p(λ) = (λ− λ0)a t(λ− λ0),

wherea = (rj − ρ)nj + rj+1nj+1 + . . . + rqnq (25)

and t(λ − λ0) is a monic polynomial of degree n − a. Moreover theconstant coefficient of t is

t(0) = (−1)m+n−a C0 det(QA P − λ0I), (26)

where Q, P are as in (6) and C0 is the sum of all principal minors ofΦj corresponding to submatrices of dimension m containing the upper

26

left block Φj−1 of Φj (if j = 1, all principal minors of dimension mare to be considered). If, in particular, m = fj for some j ∈ {1, . . . , q},then C0 is simply detΦj .

Proof: This proof resembles that of Theorem 1, but is more involved,since B is not a small perturbation of A. Thus, the simplifying assump-tion that A has only one eigenvalue cannot be made and the block J ofthe Jordan canonical form of A containing the eigenvalues different fromλ0 plays also a role. Similarly, all four blocks of the matrix B defined in(11) have to be taken into account.

We begin by writing the characteristic polynomial of A + B as

p(λ) = det((λ− λ0)I − diag(J − λ0I, J − λ0I)− B).

For the sake of simplicity we define λ ≡ λ− λ0 and p0(λ) ≡ p(λ), so thecoefficient of λn−k in p0(λ) is (−1)k times the sum of all k-dimensionalprincipal minors of diag(J −λ0I, J −λ0I)+ B. Notice that all principalminors having more than m rows containing only elements of B are zerobecause rank(B) = rank(B) = m. This simple remark is the key toprove the Theorem. The next step is to find the largest dimension ofprincipal minors having m rows which contain only elements of B. Ifthe dimension of these minors is denoted by kmax then

p0(λ) = λn−kmax t(λ),

with t a monic polynomial of degree kmax.Let α be an index set included in {1, 2, . . . , n} and denote by (diag(J−

λ0I, J−λ0I)+B)(α, α) the principal matrix of diag(J−λ0I, J−λ0I)+Bthat lies in the rows and columns indexed by α. In order to constructthe largest principal minors having m rows with only elements of B, theset α has to be of the form

α = {i1, . . . , is, a + 1, a + 2, . . . , n} with 1 ≤ i1 < i2 < . . . < is ≤ a,

because the diagonal elements of J − λ0I + B22, which have indicesa + 1, a + 2, . . . , n, are all of them different from the diagonal elementsof B22 and the largest admissible size is desired. Suppose now thati1, . . . , is are chosen among the indices corresponding to r Jordan blocksof J − λ0I. The row with the largest index chosen from a given block,say the ib-th row, contributes to the principal minor only with elementsof B, either because it is the bottom row of the block or because ib + 1does not belong to α, and thus the element in the position (ib, ib + 1)where J − λ0I has a superdiagonal 1 is not in the minor. This imposesthe restriction r ≤ m on r. Hence, to obtain the maximum number of


elements in α, i.e kmax, the indices i1 < . . . < is have to correspond toa set of m complete largest Jordan blocks of J − λ0I. The number ofpossible choices is rj !/(ρ!(rj − ρ)!), which is simply one when m = fj .In any case

kmax = r1n1 + . . . + rj−1nj−1 + ρnj + n− a,

a = n− kmax which is equation (25).Now we prove (26). Remember that t(0) is (−1)kmax times the sum

of all kmax-dimensional principal minors of diag(J − λ0I, J − λ0I) + B.Moreover the only non-zero kmax-dimensional principal minors are ofthe kind described in the previous paragraph. Consider one of theseminors and call it M . Let us denote by 1 = j1 < j2 < . . . < jh (h =kmax − (n − a) − m) the indices of the rows of this minor where J −λ0I has superdiagonal 1s. The jk-th row of this minor is the sum oftwo rows: one is the jk + 1-th row ejk+1 of the identity matrix, theother a piece of a row of B. Using this fact, we can expand M asa sum of determinants whose jk-th row is either ejk+1 or a row withonly elements of B. With the exception of the determinant with all thevectors ej1+1, ej2+1, . . . , ejh+1, the rest of these determinants are zerobecause each contains more than m rows with elements of B. A similarargument on the last n−a rows of M allows us to replace every elementof B in these rows by zero without changing the value of M . The cofactorexpansion of the remaining determinant along the rows 1 = j1 < j2 <. . . < jh leads to a value for M equal to (−1)h det(J − λ0I) times aminor of Φj corresponding to a submatrix of dimension m containingthe upper left block Φj−1. Extending this argument to all non-zerokmax-dimensional principal minors of diag(J − λ0I, J − λ0I) + B leadsto (26).

Notice that Theorem 4 is valid for any matrix B of rank m. If thefollowing two additional restrictions are imposed on B: t(0) 6= 0 andrank(A − λ0I + B) = rank(A − λ0I) + m, then A + B has m Jordanblocks less than A for λ0, and the sum of the dimensions of the remainingblocks is precisely the sum of the dimensions of the g − m smallestJordan blocks of A for λ0. It only remains to prove that changes betweenthe dimensions of theses smallest blocks do not happen. This seemsintuitively clear because the rank of B has already been used in imposingt(0) 6= 0. More on this will be said in [17].

In any case, the question raised in this section is, as far as we know,a new one in the literature. The only two references we are aware of[31, 32] are still unpublished work. In [31] the problem of rank one

28

perturbations is adressed and it is proved that the condition t(0) 6= 0 isnecessary and sufficient for A + B to have one Jordan block less thanA for λ0, and for the dimensions of the remaining blocks to be preciselythe dimensions of the g−1 smallest Jordan blocks of A for λ0. A similarbehavior appears when adding one new row and one new column to agiven matrix. This has been studied in [32, Section 2].

4. First order perturbation bounds for singularvalues

Given an arbitrary matrix A ∈ Cm×n, one can take advantage of itssingular values being eigenvalues of an associated Hermitian matrix inorder to obtain first order singular value perturbation results via Theo-rem 2. The key is using the so-called Jordan-Wielandt matrix

C =[

0 AA∗ 0

]∈ C(m+n)×(m+n) (27)

associated with A. One can easily check [35, § I.4.1] that if m ≥ n andA = UΣV ∗ is a singular value decomposition with Σ = diag(σ1, . . . , σn) ∈Rm×n, then the Hermitian matrix C has 2n eigenvalues ±σ1, . . . ,±σn

with corresponding normalized eigenvectors

1√2

[ui

±vi

],

where ui is the i-th column of U and vi the i-th column of V. In addi-tion, C has m−n zero eigenvalues with eigenvectors

[uT

i 0]T

, i =n + 1, . . . ,m.

If the matrix A is perturbed to A(ε) = A + ε B as in section 3,the Jordan-Wielandt matrix C is correspondingly perturbed to C(ε) =C + εD with

D =[

0 BB∗ 0

], (28)

i.e. to the Jordan-Wielandt matrix of A(ε) provided the perturbationparameter ε is real. Recall that both C and D are Hermitian and,consequently, normal, so according to our remark right after the proofof Theorem 2, the eigenvalues of C(ε) are analytic expansions of ε ofthe form (14) with nj = 1. Of course, this is not necessarily true for thesingular values of A(ε), due to the nonnegativity restriction. However,the loss of analyticity can only be caused by some transversal crossingσ(0) = 0 of the two eigenvalues λ(ε) = ±σ(ε) of C(ε) through theε-axis at ε = 0. Thus, to recover the singular values all we have to dois to take the nonnegative branch. In other words, to obtain the leading


term of a singular value expansion of A(ε) we compute the expansionof the eigenvalue λ(ε) = ξ ε + O(ε2) of C(ε). Then, the correspondingsingular value of A(ε) is just σ(ε) = |ξ|ε + O(ε2).

Some results on singular value perturbation expansions have been ob-tained by Stewart [34] via the ε-expansions for the eigenvalues σ(ε)2 ofA(ε)∗A(ε). Sun, on the other hand, deals in [36] with the case of sim-ple nonzero singular values, while the case of zero and multiple singularvalues is treated in [37]. Both cases are completely described in thefollowing theorem, which, to our knowledge, is new.

Theorem 5 Let A ∈ Cm×n, m ≥ n be a matrix of rank r, and let σ0

be a singular value of A.If σ0 > 0, let U0 ∈ Cm×k and V0 ∈ Cn×k be matrices whose columns

span simultaneous bases of the respective left and right singular subspacesof A associated with σ0. Then, for any B ∈ Cm×n, the matrix A(ε) =A + εB has k singular values analytic in ε which can be expanded as

σj(ε) = σ0 + ξj ε + O(ε2), (29)

where the ξj , j = 1, . . . , k are the eigenvalues of the matrix

12(U∗

0 BV0 + V ∗0 B∗U0).

If σ0 = 0 let

A =

Ur U0 Uz

Σr 00 00 0

[V ∗

r

V ∗0

]

be a singular value decomposition of A with Ur ∈ Cm×r, U0 ∈ Cm×(n−r),Uz ∈ Cm×(m−n), Vr ∈ Cn×r and V0 ∈ Cn×(n−r). Then, for any B ∈Cm×n, the matrix A(ε) = A+ εB has n− r singular values analytic inε which can be expanded as

σj(ε) = ξj ε + O(ε2), (30)

where the ξj , j = 1, . . . , k are the singular values of the (m−r)×(n−r)matrix [

U∗0

U∗z

]B V0.

Proof: As previously observed, we view the singular values of A(ε) =A + εB as the nonnegative eigenvalues of its Jordan-Wielandt matrixC(ε) = C + εD with C and D given, respectively, by (27) and (28).

30

In the simplest case when the singular value σ0 is not zero, the columnsof

1√2

[U0

V0

]∈ C(m+n)×k

form an orthonormal basis of the space of eigenvectors associated withthe semisimple eigenvalue σ0 of the Hermitian matrix C. Hence, formula(17) with Y1 = X∗

1 applied to the perturbed matrix C + εD leads toexpansion (29).

The situation is slightly more complicated when σ0 = 0, since onehas to keep track of the m − n additional null eigenvalues of C: themultiplicity of zero as an eigenvalue of C is now m − n + 2(n − r) =m + n− 2r, and the columns of

Z =1√2

[U0 U0

√2Uz

V0 −V0 0

]∈ C(m+n)×(m+n−2r)

form an orthonormal basis of the null space of C. Again, applying The-orem 2 to the perturbation C + εD leads to the expansion (30), wherethe ξj are the nonnegative eigenvalues of the matrix

M = Z∗[

0 BB∗ 0

]Z

=12

M0 + M∗0 M0 −M∗

0

√2Mz

M∗0 −M0 −(M0 + M∗

0 ) −√2Mz√2M∗

z −√2M∗z 0

,

(31)

with M0 = V ∗0 B∗U0 ∈ C(n−r)×(n−r) and Mz = V ∗

0 B∗Uz ∈ C(n−r)×(m−n).The proof is completed once we realize that the matrix M in (31) isunitarily similar to the Jordan-Wielandt matrix of

M =[

M0 Mz

]= V ∗

0 B∗ [U0 Uz

].

One can easily check that the unitary matrix

Q =1√2

In−r In−r 0−In−r In−r 0

0 0√

2Im−n

satisfies that

Q∗MQ =

[0 M

M∗ 0

],

i.e. m− n eigenvalues of M are zero, and the remaining 2(n− r) onesare plus/minus the n− r singular values of M .


5. First order perturbation bounds forgeneralized eigenvalue problems

In this final section we will just hint the close connections of the resultsin section 3 with their natural extension for generalized eigenvalue prob-lems obtained by Najman [18, 19], and Langer and Najman [10, 11, 12]in a series of papers devoted to first order eigenvalue perturbation theoryfor perturbed analytic matrix functions. We focus on these results notonly because of their great generality, but also because the main tool intheir proofs is the Newton diagram technique, the common umbrella formost of the results in the present survey. This reinforces the resemblanceof both the content of the results and the leading ideas in their proofs.

Following the presentation of Langer and Najman, we will analyze thebehavior of the eigenvalues λ(ε) of a square n× n matrix function

A(λ) + B(λ, ε) (32)

for small ε in the neighborhood of an eigenvalue λ0 of the unperturbedmatrix function A(λ), i.e. a value such that det A(λ0) = 0. The matrixfunction is assumed to be analytic around λ0 with det A(λ) 6≡ 0. Theperturbation B(λ, ε) is assumed to be analytic in a neighborhood of(λ0, 0) with B(λ, 0) = 0 for every λ.

This general framework includes most of the usual spectral problems:taking A(λ) = A − λI, B(λ, ε) = εB leads to the standard eigenvalueproblem, the choice A(λ) = λ2M + λC + K with Hermitian positivedefinite matrices M, K and positive semidefinite C corresponding toperturbed quadratic matrix polynomials appearing in vibrational sys-tems. Many other perturbed generalized eigenvalue problems can befitted into this framework.

Najman began studying in [18] the case when A(λ) is Hermitianand B(λ, ε) = εB(λ), not necessarily Hermitian. His results were anextension of those in Gohberg, Lancaster and Rodman [5] for the casewhen both A(λ) and B(λ) are Hermitian and B(λ) is positive definite.The results in [18] were precised and improved in [10], which alreadydealt with the general formulation (32). Reference [10] describes thegeneric behavior of the eigenvalues of (32) in the sense of Theorem 2above, i.e. the behavior which is to be expected unless the perturbationlies on a certain algebraic manifold of zero measure related both with theperturbation and the spectral structure of A. Further work was devotedto exploring nongeneric cases [11, 12], as well as to the study of certainspecific quadratic matrix polynomials appearing in damped vibrationalsystems [13, 19].

32

We will only state the analogous of Lidskii’s Theorem 2 as statedin [11]. As in section 3, this requires some preliminary notations. Alsoas in section 3 we simplify the presentation by assuming henceforth thatλ0 = 0 : let A(λ) be an analytic n×n matrix function with det A(0) = 0and det A(λ) 6≡ 0. The geometric multiplicity of λ0 = 0 is

g = dim ker A(0).

To define its partial multiplicities we make use of the Smith local form:one can show (see [20]) that there exist n×n matrix functions E(λ), F (λ)analytic and invertible close to λ0 = 0 such that

A(λ) = E(λ)D(λ)F (λ), (33)

with D(λ) = diag(λν1 , . . . , λνn). Notice that since the ranks of A(0)and D(0) coincide, n − g exponents νi must be zero. With no loss ofgenerality we may assume that

D(λ) = diag(λn1 , . . . , λn1 , λn2 , . . . , λn2 , . . . , λnq , . . . , λnq , 1, . . . , 1),

where n1 < n2 < . . . < nq, each exponent nj is repeated rj times forj = 1, . . . , q, and r1 + . . .+rq = g, the geometric multiplicity of λ0 = 0.The exponents nj are called the partial multiplicities for λ0 = 0 and

a =q∑

j=1

rjnj

is its algebraic multiplicity as an eigenvalue of A(λ). As in section 3, wedefine the auxiliary quantities

fj = r1 + . . . + rj , j = 1, . . . , q.

Now, consider the perturbed matrix function A(λ)+B(λ, ε), with B(λ, ε)analytic around (0, 0) and B(λ, 0) ≡ 0. Then , using (33), λ(ε) isan eigenvalue of A(λ) + B(λ, ε) if and only if it is an eigenvalue ofD(λ) + B(λ, ε) for B(λ, ε) = E(λ)−1B(λ, ε)F (λ)−1. Now, partition

D(λ) =[

D1(λ) 00 I

], B(λ, ε) =

[B11(λ, ε) B12(λ, ε)B21(λ, ε) B22(λ, ε)

],

where both D1(λ) and B11(λ, ε) are g × g, and denote

H =∂B11

∂ε(0, 0). (34)


For each j = 1, . . . , q, k = 1, . . . , rj − 1 we define ∆jk as the sumof all the (g − fj + k)-dimensional principal minors of the submatrixH(αj−1, αj−1) of H which contain the submatrix H(αj , αj), where

αj−1 = {fj−1 + 1, fj−1 + 2, . . . , g}, αj = {fj + 1, fj + 2, . . . , g}and we have used the same notation as in the proof of Theorem 4 torepresent submatrices. We also define

∆j = det H(αj−1, αj−1)

for each j = 1, . . . , q, and for convenience we set ∆q+1 = 1.

Theorem 6 (Langer & Najman [11]) If for some j ∈ {1, . . . , q} both∆j and ∆j+1 do not vanish, then there are rjnj eigenvalues of theperturbed matrix polynomial A(λ) + B(λ, ε) near λ0 = 0 admitting afirst-order expansion

λklj (ε) = (ξk

j )1/nj ε1/nj + o(ε1/nj ) (35)

for k = 1, . . . , rj , l = 1, . . . , nj , where the ξkj , k = 1, . . . , rj , are the

roots of the equation

∆j +rj−1∑

k=1

∆jkξk + ∆j+1ξ

rj = 0. (36)

The resemblance of Theorems 2 and 6 is obvious. Disregarding somenotational changes, like the partial multiplicities nj being ordered in-creasingly, or the shift of one in the subindex j, it is clear that thequantities ∆j play exactly the same role in this case as det Φj−1 in sec-tion 3. In particular, the generic behavior corresponds to nonvanishing∆j , j = 1, . . . , q. The expansion (35) is exactly the same as (12), andthe equation (36) determining the leading coefficients is the analogue of(15) or, equivalently, of (13) in this more general context. The surprisingthing is that essentially the same result is obtained in this much moregeneral context without much additional complication, just by replacingthe Jordan canonical form with the Smith local form.

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Birkhauser, Basel, 1985.

34

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